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Error Forgetting of Bregman Iteration

Wotao Yin(wotao.yin***at***rice.edu)
Stanley Osher(sjo***at***math.ucla.edu)

Abstract: This short article analyzes an interesting property of the Bregman iterative procedure, which is equivalent to the augmented Lagrangian method after a change of variable, for minimizing a convex piece-wise linear function J(x) subject to linear constraints Ax=b. The procedure obtains its solution by solving a sequence of unconstrained subproblems of minimizing J(x)+1/2||Ax-b^k||^2, where b^k is iteratively updated. In practice, the subproblem at each iteration is solved at a relatively low accuracy. Let w^k denote the error introduced by early stopping a subproblem solver at iteration k. We show that if all w^k are sufficiently small so that Bregman iteration enters the optimal face, then while on the optimal face, Bregman iteration enjoys an interesting error-forgetting property: the distance between the current point and the optimal solution set is bounded by ||w^{k+1}-w^k||, independent of all the previous errors. This property partially explains why the Bregman iterative procedure works well for sparse optimization and, in particular, for l1-minimization. The error-forgetting property is unique to $J(x)$ that is a piece-wise linear function (also known as a polyhedral function), and the results of this article appear to be new to the literature of the augmented Lagrangian method.

Keywords: Bregman iteration, error forgetting, sparse optimization, l1 minimization, piece-wise linear function, polyhedral function

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: Rice CAAM Report TR12-03, 2012

Download: [PDF]

Entry Submitted: 05/01/2012
Entry Accepted: 05/02/2012
Entry Last Modified: 05/01/2012

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